PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 11, Pages 3361–3367 S 0002-9939(00)05433-2 Article electronically published on May 18, 2000
SETS OF MINIMAL HAUSDORFF DIMENSION FOR QUASICONFORMAL MAPS
JEREMY T. TYSON
(Communicated by Albert Baernstein II)
Abstract. For any 1 ≤ α ≤ n, there is a compact set E ⊂ Rn of (Hausdorff) dimension α whose dimension cannot be lowered by any quasiconformal map f : Rn → Rn. We conjecture that no such set exists in the case α<1. More generally, we identify a broad class of metric spaces whose Hausdorff dimension is minimal among quasisymmetric images.
1. Introduction Quasiconformal homeomorphisms of a Euclidean space Rn, n ≥ 2, can distort the Hausdorff dimensions of subsets. While the dimensions of sets of Hausdorff dimension zero or n must be preserved, Gehring and V¨ais¨al¨a [5, pp. 505-506] n construct, for any β ∈ (0,n), a compact set Eβ ⊂ R with dim Eβ = β and, for 0 n n any β,β ∈ (0,n), a quasiconformal map f : R → R with Eβ0 = fEβ. Recently, Bishop [2] showed that the dimension of any compact set E ⊂ Rn of positive dimension can be raised arbitrarily close to n by a quasiconformal (quasisymmetric if n = 1) homeomorphism of Rn. On the other hand, for certain values of α ∈ (0,n), one can construct compact sets E ⊂ Rn of dimension α whose dimension cannot be lowered by any quasicon- formal mapping of the ambient space. For example, consider the closed unit k-cube [0, 1]k, k =1, 2,... ,n−1. These examples have Hausdorff dimension equal to their topological dimension and thus the Hausdorff dimension cannot be lowered by any homeomorphism of Rn. A different example (arising from the theory of removable sets for quasiconformal maps) was given by Bishop [1] in the case n =3,α =2. Bishop [2] asks whether there exist examples of this phenomenon for all 0 <α
Received by the editors October 15, 1998 and, in revised form, January 15, 1999. 2000 Mathematics Subject Classification. Primary 30C65; Secondary 28A78. Key words and phrases. Hausdorff dimension, quasiconformal maps, generalized modulus. The results of this paper form part of the author’s doctoral dissertation at the University of Michigan. The author was supported by a National Science Foundation Graduate Research Fellowship and a Sloan Doctoral Dissertation Fellowship.
c 2000 American Mathematical Society 3361
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3362 JEREMY T. TYSON
regularity, see section 2 (for the existence of such sets, note that the set Eβ in the above example of Gehring and V¨ais¨al¨aisβ-regular). We will prove a more general result (Theorem 3.4) using the machinery of the generalized (or discrete) modulus introduced in [9] and developed in [13]. The restriction to subsets of a Euclidean space is not necessary; we give a condition on metric spaces which is sufficient to guarantee that the Hausdorff dimension is minimal among quasisymmetric images (recall that in Rn, n ≥ 2, the classes of quasiconformal and quasisymmetric maps agree). It should be emphasized that Theorem 1.1 is not new; the result appears in works of Pansu [9] and Bourdon [3] (see, e.g., [9, Example 3.4] and [3, Example 1.7(b)]). These authors give a family of conditions (Lemma 3.9) on an arbitrary metric space which imply nontrivial lower bounds on the Hausdorff dimensions of quasisymmetric images. With an additional (weak) assumption, we show in Proposition 3.11 that the hypotheses of Theorem 3.4 are implied by the “best” condition of Bourdon and Pansu. The above-mentioned examples which illustrate Theorem 1.1 satisfy this “best” condition of Bourdon-Pansu. The restriction to dimensions larger than one is necessary in our proof, which relies on the geometric characterization of quasiconformality in terms of curve fam- ilies. We conjecture that no such example exists for dimensions strictly smaller than one, more precisely: Conjecture 1.2. Let n ≥ 1 and >0.IfacompactsetE ⊂ Rn has Hausdorff dimension strictly smaller than one, then there exists a quasiconformal (quasisym- metric if n =1)mapf : Rn → Rn so that fE has Hausdorff dimension smaller than . Note that such a set must be totally disconnected. For related results, see [12].
2. Notation and definitions In a metric space X,wedenotebyB(x, r) the closed ball with center x ∈ X and radius r>0. For Y ⊂ X,wewritediamY for the diameter of Y .IfB = B(x, r) is a ball in X,wewriteCB for the dilated ball B(x, Cr). For x ∈ X and r>0, we have diam B(x, r) ≤ 2r, but no lower bound for diam B(x, r) need hold in general. We call X (c1-)uniformly perfect,0
1 α α (2.1) r ≤ µ(B(x, r)) ≤ C0r C0 for all x ∈ X and 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SETS OF MINIMAL HAUSDORFF DIMENSION 3363 We call a homeomorphism f : X → Y between metric spaces (η-)quasisymmetric, where η :[0, ∞) → [0, ∞) is a homeomorphism, if (2.2) |x − y|≤t |x − z|⇒|f(x) − f(y)|≤η(t) |f(x) − f(z)| for any x, y, z ∈ X and t>0 [11]. By making η larger if necessary, we can (and will) always assume that η(t) ≥ t for t>0 and that both f and f −1 satisfy (2.2). If X → 0 is c1-uniformly perfect and f : X Y is η-quasisymmetric, then Y is c1-uniformly 0 0 perfect for some c1 = c1(c1,η) > 0. The Hausdorff dimension of a metric space is not a quasisymmetric invariant. Following Pansu [9, D´efinition 3.1], we define the conformal dimension of a metric space X to be C dim X =infdim Y, Y where the infimum is taken over all spaces Y which are quasisymmetrically equiv- alent to X. For any space X, C dim X ≥ dimT X, where dimT X denotes the topological dimension of X [7]. 3. The main result Let (X, µ) be a metric measure space. For a family Γ of curves in X and p ≥ 1, we denote by Modp Γthep-modulus of Γ [14, §6.1], [6, §2.3]. (Throughout this paper, we restrict ourselves to consider only curves whose parametrization is one-to-one.) Definition 3.1. We say that X has nontrivial p-modulus, p ≥ 1, if Modp Γ > 0for some family of curves Γ in X. Example 3.2. Let α>1. Let Z be any compact (α − 1)-regular metric space and let X be the product space Z ×[0, 1] (endowed with any of its standard bi-Lipschitz equivalent metrics). Then X is a compact, α-regular space and the family of curves Γ={γz :[0, 1] → X|γz(t)=(z,t),z ∈ Z} has positive α-modulus. Note that for any n ≥ 2and1<α≤ n,wemaychoose Z ⊂ Rn−1 (see the introduction) and so X ⊂ Rn. When 1 <α<2, the preceding example is not connected. We next give con- nected examples for each 1 <α≤ n. Example 3.3. Let 1 <α≤ n and choose a compact (α − 1)-regular Borel set Z ⊂ Sn−1.Letν be an (α − 1)-regular measure on Z and let X = {tz : z ∈ Z, 0 ≤ t ≤ 1}⊂Rn n be the cone on Z. Here, for z =(z1,... ,zn) ∈ R and t ≥ 0, we write tz = (tz1,... ,tzn). We define a measure µ on X as follows: for Borel sets A ⊂ [0, 1] and B ⊂ Z, define the spherical product set A · B = {tz : t ∈ A, z ∈ B} and set Z µ(A · B)=ν(B) · rn−1dr. A The collection of spherical product sets generates the Borel σ-algebra on X and µ is an α-regular Borel measure. Moreover, the family of curves Γ={γz :[1/2, 1] → X|γz(t)=tz, z ∈ Z} has positive α-modulus, by an argument similar to [14, Example 7.5]. Theorem 1.1 is a consequence of these examples and the following result. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3364 JEREMY T. TYSON Theorem 3.4. If X is a compact α-regular space with nontrivial α-modulus, α>1, then C dim X = α. To prove Theorem 3.4, we recall the theory of the generalized modulus, first introduced by Pansu in [9], [10] and developed by the author in [13]. Definition 3.5. Let X be a metric space and let p>0. For k ≥ 1, let Bk(X) denote the collection of all closed sets A ⊂ X for which B(x, r) ⊂ A ⊂ B(x, kr) S ∈ B B for some x X and r>0. We write (X)= k≥1 k(X). For a set function ϕ : B(X) → [0, ∞], we denote by Φp,k the Borel measure constructed by the Carath´eodory construction (Method II) using the set function ϕp and the family of sets Bk(X). See, for example, [8, §4.1] or [4, §2.10.1]. For l ≥ 1, we define a set functionϕ ˜l : Bl(X) → [0, ∞] as follows: letϕ ˜l(A)be the supremum of the values ϕ(A˜)overallsetsA˜ for which B(x, r) ⊂ A ⊂ A˜ ⊂ B(x, lr) for some x ∈ X and r>0. Finally, we denote by Φ˜ p,k,l the corresponding measure p constructed using the set functionϕ ˜l . Example 3.6. The most well-known example of a measure generated by the Cara- th´eodory construction is the Hausdorff measure Hp, which is obtained by using the set function ϕp,whereϕ(A)=diamA, on the family of all subsets A ⊂ X.Itis easy to verify that in this case p (3.1) Hp(E) ≤ Φp,k(E) ≤ 2 Hp(E) for all Borel sets E ⊂ X and all k ≥ 1. Moreover, if X is c1-uniformly perfect, then for this ϕ we haveϕ ˜l(A) ≤ (2l/c1)ϕ(A) for all A ∈Bl(X), l ≥ 1, and so p (3.2) Φp,k(E) ≤ Φ˜ p,k,l(E) ≤ (2l/c1) Φp,k(E) for all Borel sets E ⊂ X and l ≥ k ≥ 1. Definition 3.7. Let X be a metric space and let p>0. Let Γ be a family of subsets of X.Form ≥ 1,l≥ k ≥ 1, we define the generalized p-modulus of Γ (with data k, l, m) to be modp,k,l,m Γ=infΦ˜ p,k,l(X), where the infimum is taken over all set functions ϕ : B(X) → [0, ∞] satisfying Φ1,m(γ) ≥ 1 for all γ ∈ Γ. This concept was introduced in [9] and further developed in [13]. We recall some basic properties of the generalized modulus. For proofs of these results, see Propositions 3.24, 4.5 and 4.7 in [13]; note that the assumption of connectedness in Propositions 4.5 and 4.7 of [13] can be weakened to uniform perfectness. Proposition 3.8. (i) Let f : X → Y be η-quasisymmetric and let p>0.For k, k0,l,l0,m,m0 ≥ 1 satisfying l0 ≥ η(l), l ≥ k0 ≥ η(k) and m0 ≥ η(m) and any collection Γ of subsets of X, modp,k0 ,l,m fΓ ≤ modp,k,l0,m0 Γ. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SETS OF MINIMAL HAUSDORFF DIMENSION 3365 (ii) Let X be a locally compact α-regular space, α>1, with regularity constant C0. For any k, l, m ≥ 1 with l ≥ 10k, there is a constant C = C(C0,α,k,l) so that 1 Mod Γ ≤ mod Γ ≤ C Mod Γ C α α,k,l,m α for any curve family Γ in X. Before proving Theorem 3.4, we pause to discuss some earlier conditions which suffice to give lower bounds for conformal dimensions. Lemma 3.9 is not used in the proof of Theorem 3.4; we include it here in order to indicate the relationship between condition (3.3) and the hypothesis of nontrivial α-modulus in the α-regular setting. See Proposition 3.11. The following result is Lemme 1.6 of [3]. Note that the term “quasiconformal” in [3] corresponds to our term quasisymmetric and that for Ahlfors α-regular spaces, both the packing dimension dimP X and the Hausdorff dimension dimH X discussed in [3, §1.5] are equal to α. Lemma 3.9. Let X be a compact α-regular metric space and let β ≥ 0.Suppose that there exists a curve family Γ in X with the following properties: • there exists δ>0 so that diam γ ≥ δ for each γ ∈ Γ, • there exists C1 < ∞ and a probability measure ν on Γ so that β (3.3) ν{γ ∈ Γ:γ ∩ B =6 ∅} ≤ C1r for each ball B in X of radius r. Then β ≤ α − 1 and α (3.4) C dim X ≥ . α − β As β ranges from 0 to α − 1, the lower bound in (3.4) ranges from 1 to α.The “best” case is β = α−1, in which case C dim X = α. The curve family Γ in Example 3.2 satisfies (3.3) with β = α − 1. The conclusion of Lemma 3.9 is nontrivial only when α/(α − β) > dimT X.We give one example which shows that this can occur even in the case β<α− 1. Example 3.10. Let X ⊂ R2 be the Sierpinski carpet, obtained from the unit square in R2 by repeated applications of the following operation: divide each of the current squares into 9 equal subsquares and remove the central subsquare. The space X is compact and α-regular, where α =log8/ log 3. Let E ⊂ R denote the standard Cantor set and let ν be a β-regular probability measure on E, β = log 2/ log 3. Since E × [0, 1] ⊂ X, we may define a family of curves Γ={γy :[0, 1] → X|γy(t)=(y,t),y ∈ E}. For each y ∈ E,diamγy = 1. Moreover, the measure ν on E induces a measure (which we continue to denote by ν) on Γ for which (3.3) is satisfied. Thus log 8/ log 3 3 C dim X ≥ = . log 8/ log 3 − log 2/ log 3 2 Note that dimT X =1. In the “best” case β = α − 1, the result of Lemma 3.9 is contained in that of Theorem 3.4, provided that Γ contains a collection of rectifiable curves with positive ν-measure. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3366 JEREMY T. TYSON Proposition 3.11. Let X be a compact α-regular space satisfying the hypotheses of Lemma 3.9 with β = α − 1 and assume that ν(Γ0) > 0,whereΓ0 ⊂ Γ denotes the collection of rectifiable curves. Then X has nontrivial α-modulus. Proof. By passing to a further subcollection if necessary, we may assume that the lengths of the curves in Γ0 are uniformly bounded, say length(γ) ≤ M for all γ ∈ Γ0. Foreachsuchcurveγ,letmγ be the normalized arc length measure ds/M on γ. Choose λ>1. If B is a ball with radius r<δ/(2λ), then no curve γ ∈ Γ0 can lie completely in λB.Thusγ ∩ B =6 ∅ implies that length(γ ∩ λB) ≥ (λ − 1)r. Hence Z 1−α mγ(γ ∩ λB) dν(γ) 0 {γ∈Γ :γ∩B=6 ∅} Z = M α−1 length(γ ∩ λB)1−α dν(γ) {γ∈Γ0:γ∩B=6 ∅} − M α 1 ≤ r1−αν{γ ∈ Γ0 : γ ∩ B =6 ∅} ≤ C0 λ − 1 0 where C depends only on M, λ, α and the constant C1 in (3.3). By Proposition 0 2.9 of [9], modα,k,l,m Γ > 0foreachk ≥ 1, l ≥ 5λk and m ≥ 1. Choosing k, l, m 0 appropriately, we conclude (by Proposition 3.8(ii)) that Modα Γ > 0. We turn now to the proof of Theorem 3.4, which relies on the following simple observation. Lemma 3.12. Let X be uniformly perfect and let Γ be a family of subsets of X with diam γ ≥ δ>0 for each γ ∈ Γ.Letα>0.Ifmodα,k,l,m Γ > 0 for some l ≥ k ≥ 1, m ≥ 1,thendim X ≥ α. Proof. Define a map ϕ : B(X) → [0, ∞]byϕ(A)=diamA/δ. By (3.1), 1 1 Φ (γ) ≥ H (γ) ≥ diam γ ≥ 1 1,m δ 1 δ for all γ ∈ Γandsoby(3.2), 0 < modα,k,l,m Γ ≤ Φ˜ α,k,l(X) ≤ CΦα,k(X) ≤ CHα(X) where C depends only on α, δ, l, and a constant c1 of uniform perfectness for X. Hence dim X ≥ α. Lemma 3.13. Let (X, µ) be a metric measure space, p>1,andletΓ be a family 0 of curves in X with Modp Γ > 0. Then there exists δ>0 andasubfamilyΓ ⊂ Γ 0 0 with Modp Γ > 0 for which diam γ ≥ δ for all γ ∈ Γ . ∈ N ∈ Proof. For each m ,letΓSm denote the collection of those γ Γforwhich ≥ ∞ diam γ 1/m.ThenΓ= m=1 Γm. By a result of Ziemer [15, Lemma 2.3], ∈ N Modp Γ=supm≥1 Modp Γm and so Modp Γm > 0forsomem . Proof of Theorem 3.4. Let X be a compact α-regular space, α>1, and let Γ be a curve family in X with Modα Γ > 0. Let f : X → Y be an η-quasisymmetric map onto a (compact) space Y . By Lemma 3.13, we may assume that diam γ ≥ δ>0 for all γ ∈ Γ. By Proposition 3.8(ii), for any k, l, m ≥ 1withl ≥ 10k,wehave modα,k,l,m Γ > 0. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SETS OF MINIMAL HAUSDORFF DIMENSION 3367 Choosing k, l, m large enough (depending on η), we can guarantee (by Proposition 3.8(i))thatthereexistk1,l1,m1 ≥ 1with modα,k1,l1,m1 fΓ > 0. Since X and Y are compact and diam γ ≥ δ>0 for all γ ∈ Γ, we have diam fγ ≥ δ0 for all γ ∈ Γ, where δ0 = δ0(η, δ) > 0 [11, Theorem 2.5]. Finally, Y (as a quasisymmetric image of a uniformly perfect space) is also uniformly perfect, and so, by Lemma 3.12, dim Y ≥ α. The proof is complete. References [1] C. J. Bishop. Non-removable sets for quasiconformal and locally bi-Lipschitz mappings in 3 R .preprint. [2] C. J. Bishop. Quasiconformal mappings which increase dimension. Ann. Acad. Sci. Fenn. Ser. A I Math. to appear. [3] M. Bourdon. Au bord de certains poly`edres hyperboliques. Ann. Inst. Fourier (Grenoble), 45(1):119–141, 1995. MR 96b:20045 [4] H. Federer. 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MR 40:346 Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 E-mail address: [email protected] Current address: Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794-3651 E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use